The model presented here can be used to test ideas about pedaling a bike. It maintains geometrical relationships, gives plots of forces acting at the pedals, gives plots of forces powering the legs, and animates a model of your pedaling. However, the best virtual world won't turn your pedals.

Referring to Figure 1, the thigh, shin, and, crank length together with the position of the point of rotation of the thigh, point d in figure 1, define the geometry of the model. (Point d is not the height of the saddle, but the point where the thigh rotates, the point adjusted by the height of the saddle.)

The Model 

The model divides a full rotation of the pedals into a number of segments for analysis. The shaded area in Figure 1 represents the work performed by one leg moving through one of these segments. Summing over all segments (the area under the curve) gives the Work of Revolution performed by one leg in one revolution of the pedals. Work of Revolution at a given cadence is related to output power. The area under the Work-of-Revolution curve shows where in the pedal stroke work is done. Figure 1 show a representation of the output from the Model.  It shows the force vectors acting at the pedal: the tangent force at the crank circle is in green and the radial force at the crank circle is in red. The green arcs on the thigh and shin represent the strength and direction of the muscles moving the thighs and shins (labeled thigh and shin moments) and are shown to scale in model output.

Powering the Pedals, Powering the Model 

A torque is a tendency to rotate.  Muscles rotate the thighs about the hip and rotate the shin about the knee.  The strength and speed of the motions of the thighs and shins  power the pedals.  Thighs and shins can move in two directions.  Muscles that bend the thigh and shin relative to the torso are called thigh and shin flexors.  Muscles that move the thighs and shins in the opposite direction are called thigh and shin extensors. The torque produced by the thigh or shin can be different at each point in the range of motion and depends on the speed of motion.  The torque at each point is described by a function here called a strength function.  The strength and direction of the thigh and shin extensors are represented by green arcs on the thigh and shin in Figure 1.   (Some readers may think of these strength functions as strength curves or moment functions or couples.)

Figure 2, A sample plot of Thigh Extensor and Flexor Strength Functions. These functions are active, in this example, over a range of motion of about 20 to 60 degrees and show Thigh Extensor with a maximum value near 240 N m and a Thigh Flexor with a maximum value of 20 N m but in a direction of motion (negative direction) opposite to the Extensor. One can see from the example plot in Figure 2 that "pushing," (thigh extensor) has a greater torque than "pulling," (thigh flexor motion), an order of magnitude difference.

Strength Functions 

Strength functions power the model. The geometry defines the range of motions through which these functions act. The functions describe the torque (force x distance) available from the muscles at each point in a range of motion. The shape of the strength function describes how much and how quick a rider can put power into the pedals.  

In concept, a rider can measure torques using testing machines available at sports medicine clinics. Each of the four motions should be isolated for each leg, and measurements should be taken at speeds of motion corresponding to specific cadences under study. Measurements should be taken for ranges of motion specific to the geometry. The model can furnish these ranges of motion for use in collection of torque data. The dots in the example plot of Thigh Moment Functions are experimental data values. The curves are strength functions fitted by the model to these data.

Figure 3. The shape of the strength function and its maximum value depend on cadence. At fast cadence, one could expect the maximum to be less and to have a shape different from functions at slower cadences. What are the optimum cadence and strength functions to maximize power? Can riders be trained to improve their strength functions? During a sprint one could theorize that the strength function for the thigh extensor becomes very pointed and narrow (an example shown in Figure 3). What is the power from such a curve? Different geometries can change the range of motion of the strength functions and their shapes.  

"So, if one works one's quads very hard in the gym how much more power can be expected next season?" Said another way, "If one improves the quads (shin extensor strength function) by 50%, how is power affected?"

Parameters Calculated by Pedal Model 

The Pedal Model calculates the torque at the bottom bracket. This torque integrated (in the case of the model, summed over a the number of segments of rotation) over one complete rotation yields the Work of Revolution. For a given cadence, power in watts is calculated from the Work of Revolution. Plots of radial and tangential forces at the pedal, torques about the bottom bracket, and Work of Revolution are provided. Thigh and shin strength functions are plotted. An animation of pedaling showing forces and strength functions is provided.

Uses of Model 

The model can be used to test the effects of changes in position, crank length, and shin length (heel up or down can effectively change "shin length"). It can be used to test the effects of different strength functions (better pedaling techniques, stronger muscle development). How do longer cranks change power output? How is power affected by changing seat position? Do people with shorter shins have an advantage? Different shapes of the strength functions produce different power outputs. A wider range of motion could give more power. Different regions within the full range of motion could yield different power outputs. 

Special Notes 

The "shin length" is taken to be the distance from the point of rotation of the knee to the center of the pedal axle. "Shin length" can thus be changed by pointing the heel up or down. Is there an advantage to keeping one's heel down?  

The angle between the pelvis and horizontal (difficult to measure) combined with the angle between the horizontal and the thigh (easy to measure) define the range of motion of the thigh. The ranges of thigh motion given by the Pedal Model are the angle between the thigh and horizontal. This is not the same as the angle between the thigh and some point on the pelvis. It's this angle between some point on the pelvis and the thigh that one needs to measure if one expects to do weight training or collect thigh strength function data in the same range of motion as one pedals.  

The point of rotation of the thighs at the hip (see point d in Figure 1) is not the same as saddle height; although, this point of rotation can be adjusted by changing the saddle height.  

The model presented here neglects the weight of the legs. The Moment Functions power the model, and this power is used to propel a rider and bike and to also move the legs. The movement of the legs would seem to be a significant part of the total effort. Whether Moment Functions work against torque at the bottom bracket or the weight of the legs, they are still doing the work required to move the rider down the road. Measuring the Moment Functions as independent components and using the pedal model to analyze alternatives should give ways of producing maximum power from the pedals.